Abstract
In this paper, we investigate a diffusive predator-prey model with a general predator functional response. We show that there exist no nonconstant positive steady states when the interaction between the predator and prey is strong. This result implies that the global bifurcating branches of steady state solutions are bounded loops for a predator-prey model with Holling type Ⅲ functional response.
Highlights
Since the pioneering work of Holling [9, 10], the predator functional responses, especially the Holling type II functional response, have been investigated extensively
The method used here is motivated by [19], and we find that the result for the case of r = 1 in [19] can be extended to the case of r > 1 or a more general predator functional response, which satisfies Eq (3.2)
Based on Theorem 2.4 and 2.5, we have the following result on the nonexistence of nonconstant positive solutions of system (2.2) for small ρ (or equivalently, large m for system (2.1))
Summary
Since the pioneering work of Holling [9, 10], the predator functional responses, especially the Holling type II functional response, have been investigated extensively. Our result supplements the results in [27] and implies that each global bifurcating branch of steady state solutions of model (1.3) obtained in [27] is a bounded loop, which connects at least two different bifurcation points, (see Section 3). There exists a positive constant C, determined only by q, d and Ω, such that z q ≤ C inf z.
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